By Sophia Reiner
Introduction
What do flower petals, shells, and societally decided beautiful faces have in common? They fit the golden ratio. The golden ratio, also known as the Divine Ratio and Phi (φ), is a mathematical ratio existent throughout the universe. Labelled by astrophysicist Mario Livio, the golden ratio “provides us with an intriguing mingling” of the definitions of proportion “that, while defined mathematically, it is claimed to have pleasingly harmonious qualities.” However, to break this down, Euclid – the ancient Greek mathematician – told the golden ratio to be demonstrated when “the whole line is to the greater segment, so is the greater to the lesser.” What this means is that if the line below is used as an example, if the ratio of length AC is to CB the same as AB to AC, then this demonstrates the golden ratio (Livio, 2002).
When mathematically put, the proportions of AC/AB = AB/CB cross multiplied becomes x² – x – 1; this then creates the solutions of (1+√5)/2 and (1-√5)/2. The golden ratio Phi is the only number where the following can occur: “φ*φ'=-1 where φ' is the negative solution to the quadratic (1-√5)/2” (Fett, 2006; Huntley, 1970).
Geometrically, the image below – the Golden Spiral – is a logarithmic spiral with a growth factor of the golden ratio.
The golden ratio can further be quantified in the Fibonacci sequence, in which as the numbers increase, the ratio between each pair of numbers gets closer to the golden ratio (or 1.618033988749895). The Fibonacci sequence which can start with 0 or 1 follows the following sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…. ∞
The formula for this is as follows:
Fn = F(n-1) + F(n-2)
Overall, this phenomenon of the golden ratio, golden spiral, and Fibonacci sequence is relevant in human biology, desired human characteristics, and throughout nature.
The Golden Ratio in Human Biology
The golden ratio is found through the human body, such as in clenched fists, the human skull, and the heart – despite being less widely discussed. For instance, Dr. William Littler in 1973 proposed that a clenched fist can demonstrate approximately the Golden Spiral (Figure 1) (Persaud, 2015).
In regards to skull shapes, Johns Hopkins researchers Rafael Tamargo and Jonathan Pindrik identified that the dimensions of human skulls, in comparison to other animals’ skulls, display the golden ratio.
Dr Michael Henein and colleagues of Sweden’s Umea University have found the golden ratio in the hearts as well. In both healthy individuals as well as those with mild heart failure, parts of the heart – vertical and transverse dimensions – fulfill the golden ratio regardless of ethnically influenced dimensions. Furthermore, healthy ventricles’ mitral annulus (the junction between the ventricle and left atrium), the angles between the “mid-luminal axes of the pulmonary trunk and the ascending aortic continuation,” and also between the “outflow tract axis and continuation of the inflow tract axis of the right ventricle” demonstrate this golden ratio (Henein et al., 2011).
In short, Littler, Tamargo, Pindrik, and Henein support the findings that the golden ratio is naturally apparent all around the body – in bones and organs (including sensory organs).
The Golden Ratio in Surgical Interventions
Beyond the golden ratio being found throughout human biology, society has led individuals to further desire the golden ratio to be illustrated within physical characteristics. Golden ratio proportions have been consistently proposed as a method of “defin[ing] anatomical beauty” – with it being “commonly accepted that facial beauty correlates with anatomical symmetry” (Persaud, 2015; Rhodes et al., 1998). In dentistry for instance, Levin (1978) found the width of the maxillary central incisor and the width of the lateral incisor create the golden ratio proportion, as do the width of the maxillary lateral incisor with the width of the canine. This inspired later research on this golden proportion to identify the appropriate spacing of teeth, which then informed treatment; despite this, the application in creating the perfect smile has recently been questioned.
Furthermore, plastic surgery has focused on utilising the golden ratio to create ‘perfection.’ A few examples of creating this facial symmetry through cosmetic surgery include implementing the golden ratio into the ratios between the mouth and nose widths, space in the middle of each eyebrow and eyeball, and lower lip and upper lip widths (Inanli, 2021). To analyse another instance more in-depth, in abdominoplasties (known informally as a tummy tuck), the golden ratio is taken advantage of to get the goal proportion between the perimeter of the waist and the hips (Hwang & Park, 2021). Dr. Giuseppe Visconti and colleagues created a survey which presented participants with pictures of varying navel shapes and positions, in which most participants identified the photos in which the navel was moved based on the golden ratio as the most attractive – concluding that the most aesthetically attractive naval position to be at the “xiphoid-umbilicus:umbilicus-abdominal crease golden ratio.”
In essence, researchers’ continued studies on the long-lasting topic of the golden ratio in humans have identified certain fields in which societal perceptions have also grown in favour of golden ratio proportions and thus influencing surgical interventions.
The Golden Ratio Elsewhere
However, the role of the golden ratio has further significance in other fields – for instance, where it is most recognisable: nature. The following are natural examples that exclude human characteristics:
Contour spirals of shells
Ram horns
Elephant tusks
Flower petals
DNA molecules
Spiral galaxies
Other fields such as investing employ the golden ratio, in which traders and technical analysts use the ratio to predict changes to market-driven prices. To expand, technical analysis works to identify the trading signals and price patterns of past investment data to forecast future movement. While the efficacy of technical analysis is frequently debated, the golden ratio is used to identify instances of herd behaviour (Kuepper, 2022). Essentially, the golden ratio in the form of the Fibonacci sequence is converted into percentages (38.2%, 50%, and 61.8% to “estimat[e] areas of support and resistance” in the analysis of price positions for a more accurate prediction (Kuepper, 2022).
In general, the natural occurrence of the golden ratio in life is found in many existing spirals – both living and dead!
Conclusion
In conclusion, the extensive research (both historically and in the present day) on the golden ratio demonstrates its relevance not only in naturally existing life, society’s desired traits, but also in investment analysis. This implies the significance of mathematical phenomena throughout society and in understanding the mechanisms behind the choices individuals make (perceptions of attractiveness) and how their behaviour impacts greater trends (investment prices). This sparks further questions in terms of existing mathematical principles – how can chaos theory justify the presence of the golden ratio throughout the world? How can the golden ratio be taken advantage of in decision-making beyond technical analysis? Is the golden ratio a foolproof measure of aesthetic pleasure and assumptions of how things work in life? Regardless, the identification of the golden ratio in its myriad applications is in itself, a fascinating sensation.
References
Fett, B. (2006). An in-depth investigation of the divine ratio. The Mathematics Enthusiast, 3(2), 157–175. https://doi.org/10.54870/1551-3440.1047
Henein, M. Y., Zhao, Y., Nicoll, R., Sun, L., Khir, A. W., Franklin, K., & Lindqvist, P. (2011). The human heart: Application of the golden ratio and angle. International Journal of Cardiology, 150(3), 239–242. https://doi.org/10.1016/j.ijcard.2011.05.094
Huntley, H. E. (1970). The divine proportion: A study in mathematical beauty. Dover Publications.
Inanli, S. (n.d.). The Golden Ratio and Rhinoplasty. Prof.Dr.Selçuk İnanlı. Retrieved January 29, 2023, from https://selcukinanli.com/en/article/the-golden-ratio-and-rhinoplasty#:~:text=The%20golden%20ratio%2C%20which%20is%20a%20guide%20in%20plastic%20surgery,more%20beautiful%20it%20is%20perceived
Kuepper, J. (2022, July 7). Fibonacci and the Golden Ratio: Technical Analysis to unlock markets. Investopedia. Retrieved January 29, 2023, from https://www.investopedia.com/articles/technical/04/033104.asp#toc-examples-of-the-golden-ratio
Tamargo, R. J., & Pindrik, J. A. (2019). Mammalian skull dimensions and the golden ratio (Φ). Journal of Craniofacial Surgery, 30(6), 1750–1755. https://doi.org/10.1097/scs.0000000000005610
Visconti, G., Visconti, E., Bonomo, L., & Salgarello, M. (2014). Concepts in navel aesthetic: A comprehensive surface anatomy analysis. Aesthetic Plastic Surgery, 39(1), 43–50. https://doi.org/10.1007/s00266-014-0434-z
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